1.4 ­ 1.5 inclination, slope, parallel and perpendicular.notebook August 24, 2015 1.4 Inclination and Slope 1 1.4 ­ 1.5 inclination, slope, parallel and perpendicular.notebook August 24, 2015 Pre­Requisite Knowledge: Mathematical convention from Trigonometry ­ all angles measured in the counterclockwise direction are positive. Those measured in the clockwise direction are negative. We use θ to represent an angle. The inclination of a line that intersects the x axis is the measure of the smallest nonnegative angle which the line makes with the positive end of the x axis. The inclination of a line parallel to the x axis is 0. DEFINITION The inclination of a line is always less than 180o or π radians and every line has inclination. Thus for every line 0 < θ < 180o or 0 < θ < π given that θ represents inclination. DEFINITION 2 1.4 ­ 1.5 inclination, slope, parallel and perpendicular.notebook August 24, 2015 We cannot use inclination to find a simple relationship between a lines coordinates and the inclination without having to resort to trigonometric function. Thus we have to consider another expression related to the inclination of a line, namely slope. The slope m of a line is the tangent of the inclination; thus, m = tanθ DEFINITION While it is possible for two different angles to have the same tangent, two different lines cannot have the same slope because we restricted inclination to 0 < θ < 180o. Using m = tan θ we have a special situation when θ = 90o. Because tan 90o does not exist we say that vertical lines have an inclination of 90o but no slope. It is important to note that "no slope" is different from "zero slope." For all other situations we are able to rely upon the definition of the tangent function. 3 1.4 ­ 1.5 inclination, slope, parallel and perpendicular.notebook August 24, 2015 Example 1: Derive the traditional slope formula Theorem 1.4 4 1.4 ­ 1.5 inclination, slope, parallel and perpendicular.notebook August 24, 2015 Example 2: Plot the line through (2, 1) with slope 3/2. 5 1.4 ­ 1.5 inclination, slope, parallel and perpendicular.notebook August 24, 2015 1.5 Parallel and Perpendicular Lines 6 1.4 ­ 1.5 inclination, slope, parallel and perpendicular.notebook August 24, 2015 Parallel: If two non vertical lines are parallel, then they must have the same inclination and, thus, the same slope. If two parallel lines are vertical, then neither one has slope. Similarly, if m1 = m2 or if neither line has slope, then the two lines are parallel. Thus two lines are parallel if and only if m 1 = m2 or if neither line has slope. 7 1.4 ­ 1.5 inclination, slope, parallel and perpendicular.notebook August 24, 2015 Perpendicular: If two non vertical lines l 1 and l2 with the respective inclinations θ1 and θ2 are perpendicular then o o θ1 = θ2 + 90 . Thus applying the definition of slope tan θ1 = tan (θ2 + 90 ) = ­cot( θ2) = ­ 1/(tan( θ2)). Thus m1 = ­1/m 2. 8 1.4 ­ 1.5 inclination, slope, parallel and perpendicular.notebook August 24, 2015 Theorem 1.5 9 1.4 ­ 1.5 inclination, slope, parallel and perpendicular.notebook August 24, 2015 Example 3: Determine if l 1 containing the points (1,5) and (3, 8) is parallel, coincident, perpendicular, or neither to l2 which contains the points (­4, 1) and (0, 7) 10 1.4 ­ 1.5 inclination, slope, parallel and perpendicular.notebook August 24, 2015 Example 4: If the line through (x, ­3) and (3, 1) is perpendicular to the line through (x, ­3) and (­1, ­2), find x. 11.